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University Math Lab: Systems & Structures FREE

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Taylor Series

75 min University Math Lab: Systems & Structures

Learning Objectives

  • Construct Taylor polynomials
  • Find radius of convergence
  • Apply to approximation

Key Concepts

Taylor Series

f(x) = Σ f⁽ⁿ⁾(a)/n! · (x-a)ⁿ

Maclaurin Series

Taylor series centered at a = 0

Theory

**Taylor Series centered at a:** f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)²/2! + ... **Common Series:** - eˣ = 1 + x + x²/2! + x³/3! + ... - sin(x) = x - x³/3! + x⁵/5! - ... - cos(x) = 1 - x²/2! + x⁴/4! - ...

Examples

First 3 terms of eˣ at x = 0

Solution: 1 + x + x²/2

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Knowledge Check

Test your understanding of The Taylor Sculptor!